Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 305-308
3O5
SOME FRIEDMANN COSMOLOGICAL SOLUTIONS IN THE SCALE
COVARIANT THEORY OF GRAVITATION
AROONKUMAR BEESHAM
Department of Applied Mathematics
University of Zululand
Private
Bag X1001, Nwa-Dlangezwa 3886, South Africa
(Received December 2, 1986)
ABSTRACT. The scale covariant theory of gravity admits the possibility of a time varying
gravitational constant but contains a gauge function for which there is no independent equation.
The circumstances under which explicit forms for a gauge function may be derived within the
context of Friedmann-FLobertson-Walker cosmological models are investigated and several forms
are derived.
KEY WORDS AND PI-IltASES. Alternative gravitational theories, scale covariant theory,
cosmology, Friedmann models.
1980 AMS SUBJECT CLASSIFICATION CODES. 83D, 83F.
1.
INTRODUCTION.
It is well known that general relativity does not admit a time varying gravitational
constant G. Numerous attempts to study such a variation of G within general relativity have not
proved successful [1-3]. The scale covariant theory of Canuto et al [4],is a viable scalar tensor
theory which allows for a varying (3 [5,6]. The field equations in this theory for zero cosmological
constant are
[7].
where the function
In equation (1),
ab
R.ab is the Ricci tensor, lZ the Ricci scalar and Tab the energy momenttim tensor.
We are using units in which the speed of light is unity and the constant 8x has been absorbed into
G. The gauge function fl is a scalar function satisfying 0 < < Latin indices takes values 0 to
3. A semi-colon denotes covariant derivative and we are using the notation fl for the ordinary
(R).
derivative of/3 with respect to x a. For further details of the signs and conventions we are using
see Ellis [8].
A. BEESHAM
306
A. pecular feature of the scale covariant theory is that no independent equation for t3 exists
also
[9]). The normal procedure is to dopt an equation of the form [11]
(see
fl(t)=(t It) ’, e= -1/2,l,t =constant,
(3)
O
O
and then to try to determine which form best fits the observations and experimental results.
However there are certain instances in which we may determine explicit forms for/3. We shall
investigate when this is possible within the context of Friedmann-Kobertson-Walker cosmological
models and then determine some explicit forms for
.
2.
FIELD EQUATIONS.
The universe appears spatially homogeneous and isotropic about us. This leads to the
Robertson-Walker metric which in comoving coordinates has the form
ds -dt + Ftg(t)[drg-/(1--kr-) + r-d0 + rsing-0d],
(4}
where k -1 or 0. We assume the perfect fluid form for the energy momentum tensor
Tab
(#+p)u aub + Pgab’
(5)
where # and p are, respectively, the energy density and pressure of the fluid, u is the fluid
.
ua
gab is the metric tensor. In the comoving coordinates we are using, has the
form ua
The field equations (1) and (2) become, for the Robertson-Walker metric {4)and
the energy momentum tensor (5) [4]
3[R, + 3// + 3Rfl/(R)= -,}G(#+3p),
(6)
four-velocity and
3(R,/R, + //)- + 3k/R,
G#,
(7)
where the overhead dot denotes a derivative with respect to time.
Assuming an equation of state of the form
p
"t--l)/, 1 _< ’)’ _< 2,
equations (6) and (7) lead us to the following conservation equation
This equation may readily be integrated to yield
#= t/(r’rp’r-),
f9
We have here two independent equations (7) and (9) but four unlnowns R,,/, (3 and #.
Canuto and his coworkers attempted to solve these equations by assuming equation (3) for and
also the condition G l/t, which is a consequence of Diracs large numbers hypothesis [11]. Since
observations have indicated that a variation of G 1/t is not likely (Hellings et al [12]), we do not
assume this condition. Rather we shall seek static solutions and show that they lead to explicit
forms for//. Static solutions are important in the scale covariant theory because nonstatic
Friedmann solutions in general relativity could appear to be static in the scale covariant theory
[3].
Substituting equation (9) into equation (7), we obtain
For static solutions l:t
3(P,,/P,, + ,a/, + 3kiP,? I/(P,,:Fr’r-).
0, R, RO constant, and equation (10) becomes
3/I/ + 3klR,0
We now discuss the solutions to equation (11).
-=
ll(lq,0 ’r/’r-).
(10)
(11)
FRIEDMANN COSMOLOGICAL SOLUTIONS
307
SOLUTIONS FOR k=0
The solutions can be written in the compact
(12)
SOLUTIONS FOR k=+l
Dust (v=l)
(i)
,= [].- o4t/rt )]/(6R. ).
(it)
(13)
Radiation (7=4/3)
/3- [sin(t/R
(iii)
Stiff matter (7=2)
[sin
SOLUTIONS FOR k =-1
Dust (9, 1)
(i)
/3=
(it)
(iii)
2t/(,f-R 4)]’2.
[cosh(t/Ro)]/(6FI,o)- 1/(6FI,o).
4/3)
/3
Stiff matter (7 2)
Radiation (9’
[sinh(t/Ro)]/(fR.o).
fl [sinh(2t/R )]/(3R 2).
6.
13(0)
DISCUSSION
In each of the above solutions we have chosen a constant of integration to be zero so that
has also been given by Rahman and Banerii [13]. Our
0. The solution (12) for 9’
solutions can also be obtained by transformations from the nonstatic Friedmann-l:iobertsonWalker solutions in general relativity given by Vajk [14]. (Note that the k +1 case of Vajk
corresponds to our k -1 case and vice versa). It is these nonstatic solutions which would appear
to be static in the scale covariant theory.
It is clear from our solutions and equation (9) that G# varies with time even though the
solutions are static (see also [13]). This is not possible in the case of static cosmological models in
general relativity.
The scale covariant theory is a generalization of general relativity. The procedure for
obtaining corresponding general relativistic solutions (when they exist) from those in the scale
covariant theory is to put/3 1 and G constant [4, 13]. Our solution (13) reduces under this
procedure to the static Einstein universe in general relativity (see [8] for the latter). Apart from
this specific solution, our other solutions have no corresponding static general relativistic
analogues.
Explanation of the observed red shifts of galaxies cannot be given in the conventional way
as due to recession of galaxies. However the main importance of the presented solutions is not so
much their viability but in the forms,of the gauge function.
Finally, we remark that he nonstatit-Friedmann-Fi.obertson-Walker models in the scale
covariant theory have been studied by Catiuto et al [10] (the dust and radiation cases) and by us
[15] (the stiff matter case). In these solutions the gauge function appears as an arbitrary function.
308
A. BEESHAM
and G constant, the general relativistic solutions given by Vajk which we
By putting/3
mentiondd earlier are obtained.
As far as the physical interpretation of our solutions is concerned, we note that whilst in
the gravitational units of general relativity the universe appears to be expanding, in the atomic
units of the scale covariant theory, the universe could appear to be static [13]. Furthermore, for
theoretical reasons, the study of static universes is appealing [16]. The value of the Hubble
constant has been steadily decreasing from the initial value proposed by Hubble. If an alternative
explanation for the red shift gains acceptance, static models could be back in contention [16].
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